MCDM - Multi Criteria Decision Making

MCDM - Multi Criteria Decision Making.

Darko Milosevic, PhD

Studies Using Mathematical Modeling with the MCDM Techniques

A combination of the Analytic Hierarchy Process (AHP) and Goal Programming (GP) for a global facility location-allocation model is proposed by Badri [26]. Lee and Kim studied using the ANP-GP integrated model for interdependent information system project selection, and with this model, the most appropriate project is selected from six different projects [27]. Karsak [28] studied product planning in quality function deployment using a combined ANP-GP approach. Chang et al. [29] proposed an integrated model using ANP-GP for revitalization strategies in historic transport, and with this model, they chose the most appropriate project through seven different projects for the Alishan Forest Railway. Yan et al. studied the prioritized weighted aggregation in MDCD [30]. An ANP-GP-based integrated study of the problem of resource allocation in transportation [31], a study about exams weighted with ANP [32], the applications of the multi-objective and criteria decision-making process [33], a goal programming approach for the multi-purpose and multi-choice assignment problem [34] and a study of the scheduling of classes in a school [35] can be mentioned as examples for this area.

Studies about Vehicle Selection

Byun [36] proposed the AHP approach for selecting an automobile purchase model, which may affect the vehicle selection criteria to select the best car among three different cars. A multi-criteria optimization method for the vehicle assignment problem in a bus transportation company is studied by Zak et al. [37]; the routes of the buses are found to minimize the cost of transportation. Apak et al. [38] studied an AHP approach with a novel framework for luxury car selection. Seven criteria and criteria weights are determined for the personal vehicle selection by this proposed model. Nine criteria have been identified, and eight alternatives were ranked according to their weights by Gungor and Isler [39], who studied automobile selection with the AHP approach. Selecting the best panelvan car by using the Promethee method is studied by Soba [40]; one of the six vehicles has been selected with this proposed model, and six criteria were used in the decision model.[1]

The analysis and evaluation of decisions include the process of recognizing, quantifying and comparing the benefits and costs of decision criteria.

Table 2. The fuzzy important weight, BNP and rank of each criterion

	Fuzzy importance weight Wj			BNP	Rank
C1	0,4199	0,6200	0,8066	0,6155	8
C2	0,4733	0,6733	0,8599	0,6688	5
C3	0,5666	0,7666	0,9199	0,7510	2
C4	0,4466	0,6466	0,8199	0,6377	7
C5	0,5133	0,7133	0,8799	0,7022	4
C6	0,5799	0,7799	0,9333	0,7644	1
C7	0,5399	0,7399	0,9066	0,7288	3
C8	0,4733	0,6733	0,8466	0,6644	6
C9	0,3999	0,5933	0,7733	0,5888	12
C10	0,4199	0,6199	0,7999	0,6132	9
C11	0,4066	0,6066	0,7933	0,6022	10
C12	0,4066	0,6066	0,7866	0,5999	11
C13	0,3799	0,5799	0,7799	0,5799	13
C14	0,3799	0,5800	0,7666	0,5755	14
C16	0,3799	0,5800	0,7666	0,5755	14

4.2. Construct the fuzzy decision matrix

First the evaluators adopted linguistic terms (Table 1), including ‘‘very poor’’, ‘‘poor’’, ‘‘fair’’, ‘‘good’’, ‘‘very good’’ to express their opinions about the rating of every solar cell performance criteria, listed in Table 3.

Table 3. Technological performance data of seven Polycrystalline Silicon Solar Panel type

Criteria	Panel Specifications
(Electrical Details)	JAP-6	SUN	TSM	REC255	POLY	ES-H	Hyundai
	72-320	260-60	260	PLUS	245	280	230
Module Efficiency (%)	16.51	16.02	15.9	15.5	14.6	14.31	14.2
Nominal Power (Watts)	320	260	260	255	245	280	230
Max Power (Volts)	37.62	30.60	31.30	30.60	30.60	35.71	29.40
Max System Voltage (Volts)	1000	1000	1000	1000	600	600	1000
Max Power Current (Amps)	8.51	8.50	8.31	8.34	8.02	7.87	7.90
Open Circuit Voltage (Volts)	45.98	38.00	38.20	38.00	37.50	44.06	36.90
Short Circuit Current (Amps)	8.89	9.01	9.02	8.89	8.62	8.55	8.40
Cell Configuration	72	60	60	20	60	36	60
Normal Operating Cell (NOCT)	46 °C	46 °C	46 °C	46.9 °C	45.5 °C	46 °C	46 °C
Temperature Coef. (%/°C)	-0.44	-0.50	-0.41	-0.36	-0.44	-0.43	-0.44
Temperature Coef. (%/°C)	0.06	0.06	0.05	0.02	0.04	0.05	0.05
Temperature Coef. (%/°C)	-0.34	-0.37	-0.32	-0.26	-0.32	-0.31	-0.34
Max Load Snow (Pa)	5400	5400	5400	5400	5400	3828	5400
Max Load Wind (Pa)	2400	2400	2400	2400	5400	3828	2400
Weight (Kg)	22.50	19.10	19.50	18.00	18.80	23.40	19.00

The fuzzy performance ratings of each solar cells candidate regarding evaluation criteria were averaged to synthesize the various individual judgments. With Eq. (14), the synthetic fuzzy decision matrix can be computed as in Table 4.

4.3. Normalize the fuzzy decision matrix

To ensure that the normalized triangular fuzzy numbers were included in the interval [0,1], linear scale transform functions were utilized in this study. By applying Eqs. (15) and (16), the synthetic fuzzy decision matrix were normalized, and the results are shown in Table 5.

4.4. Establish the weighted normalized fuzzy decision matrix

Since the importance weights of criteria are different, the weighted normalized fuzzy decision matrix can be obtained using Eqs. (17) and (18), and the results are presented in Table 7. For instance, consider the fuzzy numbers (0.22, 0.48, 0.81) of alternative  with respect to criterion  listed in Table 7:

0.22 = 0.52  0.42;         0.48 = 0.770.62;         0.81 = 1.00  0.81

Table 4. The fuzzy decision matrix of seven solar cells candidate regarding each criterion

Table 5. The fuzzy normalized decision matrix of seven solar cells candidate

Table 6. The fuzzy weighted normalized decision matrix of seven solar cells candidate

Table 7. The closeness coefficient and rank of seven solar cells candidate

	Solar Cells	di+	di-	CCi	Rank
A1	JAP6	8,351	8,074	0,492	1
A2	SUN	8,443	7,947	0,485	2
A3	TSM-260	8,765	7,479	0,460	3
A4	REC255	9,209	6,896	0,428	5
A5	POLY	8,960	7,214	0,446	4
A6	ESH	9,210	6,880	0,428	6
A7	HYUNDAI	9,652	6,325	0,396	7

4.5. Calculate the distance of each solar cell to FPIRP and FNIRP

Eqs. (11) and (21)–(22) respectively derive the distance of each solar cell candidate to the fuzzy positive and fuzzy negative ideal reference point, as shown in Table 8. Take  and as shown in Table 7.

4.6. Obtain the closeness coefficient for ranking of seven initial solar cells

Once the distances of solar cells from FPIRP and FNIRP are determined, the closeness coefficient can be obtained with Eq. (23). The index for second candidate solar cell is calculated as:

A solar cell candidate with a closeness coefficient close to 1 has the shortest distance from the fuzzy positive ideal reference point, and the largest distance from the fuzzy negative ideal reference point. In other words, a large closeness coefficient of solar cell indicates good performance. Table 8 shows the seven initial solar cell product in accordance with the closeness coefficient. Therefore, their ascending rank is substituted as follows: .

That is, JAP-320 (0.492) > SUN-260 (0.485) > TSM-260 (0.460) > POLY-245 (0.446) > REC-255PE (0.428) > ESH-280 (0.428) > HYUNDAI-230 (0.396).

The JAP6-320 having the largest closeness coefficient value, is the best among the even initial solar cells.

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[1] Acar, Engin, and Hakan Soner Aplak. "A Model Proposal for a Multi-Objective and Multi-Criteria Vehicle Assignment Problem: An Application for a Security Organization." Mathematical and Computational Applications 21.4 (2016): 39.